50. Call a subset S of the integers 1, 2, ..., n...

50. Call a subset S of the integers {1, 2, ..., n} extraordinary provided its smallest integer equals its size: min{x : x ? S} = |S|. For example, S = {3, 7, 8} is extraordinary. Let g_n be the number of extraordinary subsets of {1, 2, ..., n}. Prove that g_n = g_{n-1} + g_{n-2}, (n ? 3), with g_1 = 1 and g_2 = 1.

Transcript

00:01 In this question, we have an expression restricted by some conditions.

00:04 So we need to basically prove that g of n is equal to g of n minus 1 plus g of n minus 2, where n is greater than equals to 3.

00:15 Right.

00:15 So now we can say that let j of n is equal to, it will be 1 comma 2 and this go on up to n, right? and then here we can say the extraordinary subset of j of n can be divided into two parts and they will be basically a and the other one will be b and they are mutually exclusive and hence we can say that g of n will be equal to mod of a plus mod of b and hence we can say the extraordinary subsets of j of n.

00:48 We are talking about extraordinary subsets of j of n which do not contain n.

00:54 So here it does not contain n so it can be written for best.

00:59 Better understanding as well.

01:02 So it will be like this.

01:04 Precisely the extraordinary subsets of j of n minus 1 and hence we can say it will be j of n minus 1 and then here from this very information we can say g of n minus 1 will be equal to a mod of a.

01:21 Basically what we have done in the above statement we have said the extraordinary subsets of j of n which do not contain n are the extraordinary subsets of j of n minus 1.

01:33 And hence we have written this very expression.

01:36 So proceeding further, we can say extraordinary subsets of j of n which contains n.

01:43 So here we are talking about extraordinary subsets of j of n which contains n have a bijection with extraordinary subsets of j of n minus 2.

01:55 So now we can say that let x be the.

01:59 Subset of j of n minus 2.

02:03 Now x dash be the subset formed by increasing all the elements of x by 1.

02:11 So x dash is formed by increasing all the elements of x by 1 and hence we can say that call f of x and in that case it will be a set formed by adding n to x.

02:27 So f of x is formed by adding n to x.

02:29 X set...

Question

Please give Ace some feedback